MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2005 India National Olympiad
2005 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
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Find all real functions withf(x^2 + yf(z)) = xf(x) + zf(y)
Find all functions
f
:
R
⟶
R
f : \mathbb{R} \longrightarrow \mathbb{R}
f
:
R
⟶
R
such that
f
(
x
2
+
y
f
(
z
)
)
=
x
f
(
x
)
+
z
f
(
y
)
,
f(x^2 + yf(z)) = xf(x) + zf(y) ,
f
(
x
2
+
y
f
(
z
))
=
x
f
(
x
)
+
z
f
(
y
)
,
for all
x
,
y
,
z
∈
R
x, y, z \in \mathbb{R}
x
,
y
,
z
∈
R
.
5
1
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A sequence of integers has infinitely many even numbers
Let
x
1
x_1
x
1
be a given positive integer. A sequence
{
x
n
}
n
≥
1
\{x_n\}_ {n\geq 1}
{
x
n
}
n
≥
1
of positive integers is such that
x
n
x_n
x
n
, for
n
≥
2
n \geq 2
n
≥
2
, is obtained from
x
n
−
1
x_ {n-1}
x
n
−
1
by adding some nonzero digit of
x
n
−
1
x_ {n-1}
x
n
−
1
. Prove that a) the sequence contains an even term; b) the sequence contains infinitely many even terms.
4
1
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All possible 6-digit numbers are written as a sequence
All possible
6
6
6
-digit numbers, in each of which the digits occur in nonincreasing order (from left to right, e.g.
877550
877550
877550
) are written as a sequence in increasing order. Find the
2005
2005
2005
-th number in this sequence.
2
1
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\frac{43}{197} < \frac { \alpha }{ \beta } < \frac{17}{77}
Let
α
\alpha
α
and
β
\beta
β
be positive integers such that
43
197
<
α
β
<
17
77
\dfrac{43}{197} < \dfrac{ \alpha }{ \beta } < \dfrac{17}{77}
197
43
<
β
α
<
77
17
. Find the minimum possible value of
β
\beta
β
.
1
1
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M is the midpoint of the side BC of a triangle ABC
Let
M
M
M
be the midpoint of side
B
C
BC
BC
of a triangle
A
B
C
ABC
A
BC
. Let the median
A
M
AM
A
M
intersect the incircle of
A
B
C
ABC
A
BC
at
K
K
K
and
L
,
K
L,K
L
,
K
being nearer to
A
A
A
than
L
L
L
. If
A
K
=
K
L
=
L
M
AK = KL = LM
A
K
=
K
L
=
L
M
, prove that the sides of triangle
A
B
C
ABC
A
BC
are in the ratio
5
:
10
:
13
5 : 10 : 13
5
:
10
:
13
in some order.
3
1
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Three quadratic equations
Let
p
,
q
,
r
p, q, r
p
,
q
,
r
be positive real numbers, not all equal, such that some two of the equations \begin{eqnarray*} px^2 + 2qx + r &=& 0 \\ qx^2 + 2rx + p &=& 0 \\ rx^2 + 2px + q &=& 0 . \\ \end{eqnarray*} have a common root, say
α
\alpha
α
. Prove that
a
)
a)
a
)
α
\alpha
α
is real and negative;
b
)
b)
b
)
the remaining third quadratic equation has non-real roots.